This paper considers the problem of solving parameter-dependent Riccati equations. In this paper, a tractable iterative scheme involving mainly additions and multiplications is developed for finding solutions to arbitrary accuracy. It is first presented in the parameter-independent case and then extended to the parametric case. It hinges upon two results: (i) a palindromic quadratic polynomial matrix characterization of the matrix sign and square root functions. (ii) a particular representation of parameter dependent matrices with negative and positive power series with respect to parameters. Several numerical examples are given throughout the paper to prove the validity of the proposed results.